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Wikipedia / Volume conjecture

In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements.

Statement

Let O denote the unknot be the Kashaev invariant, which may be defined as

  1. ,

-Colored Jones polynomial. The volume conjecture states that

  1. ,

is the simplicial volume in the 3-sphere, defined as follows. By the JSJ decomposition may be uniquely decomposed into a system of tori

hyperbolic Seifert-fibered. The simplicial volume is then defined as the sum

  1. ,

is the hyperbolic volume.

is a hyperbolic knot.

History

The Kashaev invariant was first introduced by Rinat M. Kashaev in 1994 and 1995 for hyperbolic links as a state sum using the theory of quantum dilogarithms. Kashaev stated the formula of the volume conjecture in the case of hyperbolic knots in 1997.

pointed out that the Kashaev invariant is related to the colored Jones polynomial. They used an R-matrix as the discrete Fourier transform for the equivalence of these two descriptions. This paper was the first to state the volume conjecture in its modern form using the simplicial volume. They also prove that the volume conjecture implies the following conjecture of Victor Vasiliev:

  1. If all Vassiliev invariants of a knot agree with those of the unknot, then the knot is the unknot.

.

Status

The volume conjecture is open for general knots, and it is known to be false for arbitrary links. The volume conjecture has been verified in many special cases, including:

Relation to Chern-Simons theory

Using complexification, ,

  1. ,

is the Chern–Simons invariant. This suggests a relationship between the colored Jones polynomial and complexified Chern–Simons theory.

References

Notes

Sources

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